Date of Award


Document Type


Degree Name

Master of Arts (MA)



First Advisor

Dr. Jack Heidel


This thesis explores methods for finding cycles in synchronous Boolean networks. There is a brief survey the functional iteration method for finding cycles in a synchronous Boolean network. This method can, in theory, completely describe a Boolean network, but is inefficient and unfeasible in practice, especially for large networks. A more direct computational approach, referred to as the computational iterative method, is introduced and the results of this method as applied to three node synchronous Boolean networks are presented. This analysis reveals interesting behavior about these simple networks that in some cases can be generalized to networks of any size. The main focus of the paper is the use of scalar equations, self-referring logic equations that describe the behavior of a node, to determine the possible cycle lengths within a synchronous Boolean network. The scalar equation method is introduced and a theorem is presented that proves the existence of scalar equations contingent on the Jacobian condition, a condition required for the solvability of a system of equations. The reduced scalar equation is presented. The reduced scalar equation is a simplified form of the scalar equation that can be utilized to determine the possible cyclic structure of a network. In addition, two theorems are introduced that define a class of scalar equations that can easily be reduced to scalar form.


A Thesis Presented to the Department of Mathematics and the Faculty of the Graduate College University of Nebraska In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics University of Nebraska at Omaha. Copyright 2002 Christopher Farrow